494 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
where, as before, the upper or lower sign is taken according as the imaginary part of 
[jl is positive or negative. 
35. When p = cos 9, and 6 lies between 0 and \i r, the expression on the left-hand 
side of (68) becomes, on putting p = cos 9 + 0 . t, 
^ cog Pj — mm g * n . g’ ni,rI i Q/ 1 (cOS 0) — — . P /" (cOS 9) Y , 
and on the right-hand side (p 3 — l)* m = e imm sin® 6, hence (68) becomes 
2 
cos W7r . V n m (cos 6) — — sin m77 . Q,® (cos 6) 
II (u + to) 
~ n (n - to) ¥‘U (- i) n (to - J) 
n ^ + ^ -— sin® 6 f (cos 6 + t sin 9 cos \b) >l ® sin 2 " 1 !// dxjj . (72). 
— 4) n (TO — ' ) J n ' r i \ / 
Again, on putting p = cos 6 — 0 . t, we find in a similar manner 
cos mu . P, ® (cos 9) 
II (n + to) 
77 
sin mu . Q,® (cos #) 
1 
II (n — to) 2" i II (— |) II (to 
—— sin® 0 [ (cos 0 — t sin # cos r//)" ® sin 2 ® if/ chfj . (73). 
— 27 Jo 
Again, putting p = cos 9 A 0. t in (55), we have 
cos rrn t . P„® (cos 9) — — sin mu . Q,® (cos 9) 
II (n -f- m) sin" 1 9 C n 
— J r> I 
sill 2 '" -v/r 
IT (n — m ) 2*n (—■§•) IT (m — i) J o (p + \/p 2 — • 1 cos A)" +n,+1 
■ ( 74 )- 
Next let us consider the case in "which 9 lies between | u and u ; w-e find from 
(70), by putting /x = cos 9 A 0 . t, 
2 _ 
77 
—- [ (cos 9 A <- sin 9 cos ii/)"“® sin 2 ® xfj dip . ( 75), 
e Tmm . P ; ® (cos 9) {1 At sin n77. e ±/i,u }-— sin nu . e ±( " ® >m Q„® (cos 9) 
II (n + to) sin’" 0 
II (n — to) 2'“II (— J>) II (to 
29 - o 
this corresponds to (72), (73); the phase of cos 9 A t sin 9 cos xp, when \b = ^u, is 
A tt or — u according as the upper or the lower signs are taken in the exponentials. 
Again, corresponding to (74), we find that when 9 lies between ^77 and 77 , 
2 
— e T(u+m)nL . P,® (cos 9) {e*" m A 1 sin nu } H-sin nu . e T( " + ® },rt Q„® (cos 9) 
IT (n + to) sin" 1 6 r sin 2 " 1 yfr 
II (n — to) 2" 1 II (— A) n (?» —- A Jo ( cos 9 + t sin 6 cos -yp) n+M 
»d+ • (70). 
